According to the completeness lemma in [1]}, if the properties \(1-3\)  in Theorem 1 hold for consensus rounds \(\lbrace 0,1,\cdots ,r-1\rbrace \) , then properties 1 and 2 hold for consensus round \(r\) , when the consensus leader \(n_{l^{r}}\)  is honest. As we have discussed above, the use of the reputation values list does not have any effect on the upholding of either of the aforementioned properties. Hence the completeness will uphold for the modified consensus procedure also.
